SIGEST

Abstract

The SIGEST paper in this issue, “The Smallest Possible Iteration Radius for Synchronization of Self-Propelled Particles,” by Ge Chen, Zhixin Liu, and Lei Guo, is from the SIAM Journal on Control and Optimization. A self-propelled particle system is a collection of particles or agents which has discrete dynamics. A particle updates its velocity at each time step by a random perturbation of the average velocity of its neighbors. The question addressed in this paper is how limited the neighborhood can be before the system fails to synchronize. Self-propelled particle systems model collectives such as colonies of bacteria, herds of animals, swarms of insects, or flocks of birds. If the interacting range about a particle is large, it is easy for the system to reach an ordered state. If the interaction range is very small, so small that a particle only senses itself, the system cannot synchronize. The result in this paper is that the smallest possible interaction range is approximately $\sqrt{\log n/(\pi n)}$, where n is the number of particles in the system. The mathematics is an interesting but challenging mix of graph theory and dynamical systems. The authors have expanded the paper to make the first three sections accessible to most SIAM Review readers.